ket equilibrium conditions for lossless-DLMPs with inter- temporal energy requirements ... The convexified lossy-DLMP methods [3], [12] are more accurate than ...
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Decomposition and Equilibrium Achieving Distribution Locational Marginal Prices using Trust-Region Method Sarmad Hanif, Student Member, IEEE, Kai Zhang, Student Member, IEEE, Christoph Hackl, Senior Member, IEEE, Masoud Barati, Senior Member, IEEE, Hoay Beng Gooi Senior Member, IEEE, Thomas Hamacher
Abstract—We propose a new distribution locational marginal price (DLMP) model which is based on a linearized variant of the global energy balance formulation along with trustregion based solution methodology. Compared to existing DLMP works in the literature, the proposed DLMP model has shown to depict the following features: i) It decomposes into most general components, i.e., energy, loss, congestion and voltage; ii) it presents market equilibrium conditions; and ii) it is capable of achieving an efficient flexibility resource allocation in local dayahead distribution grid markets. The developed model is tested first on a benchmark IEEE 33-bus distribution grid and then on much larger grids with the inclusion of dispatch from flexible loads (FLs) and distributed generators (DGs). Index Terms—Distribution Locational Marginal Prices (DLMPs), Linearization, Market Equilibrium, TrustRegion, Distributed Generators (DGs), Flexible Loads (FLs)
I. I NTRODUCTION The idea of establishing DLMPs has started to gain attention in the power system community [1]–[3]. Essentially, DLMPs are distribution grid variants of locational marginal prices (LMPs) at the transmission grid level [4] which provide the theoretical foundation to form deregulated energy markets around the world. As contemporary power systems are increasingly facing the integration of DGs and FLs at distribution grids, transmission level (wholesale) markets might not be sufficient for achieving economic operation in power systems. As reported in [2], [5]–[9], if not controlled properly, DGs and FLs cause congestion and higher losses in distribution grid operation. As a remedy, DLMPs are proposed as the mechanism to provide economical price signals in distribution grids, with applications ranging from loss reduction [2], [10], higher local DGs and FLs utilization [1], [10] to congestion management in distribution grids [5]–[8]. One of the major challenges, which is also the main focus of this paper is the calculation of DLMPs, that not only achieves overall grid social welfare maximization but also for local flexibility resources, with consideration of nonlinear distribution grid power flows and inter-temporal energy requirements. A. Related Work Almost all recent literature on DLMPs [1]–[3], [5]–[8], [10] makes the distribution service operator (DSO) responsible for running the local market with aim of maximizing the social welfare of its underlying grid. On the other hand, flexibility resources (FLs and DGs) maximize their individual surplus. The coordination between FLs/DGs and the DSO is then established to find DLMPs, maximizing the individual as well as the overall surplus of the grid, i.e. a unique dispatch for FLs and DGs. Based on the formulation of DLMPs, two broad methods, (i) lossless-DLMPs and (ii) lossy-DLMPs exist.
Recent lossless-DLMP formulations are reported in [5]– [8]. These works presented solution uniqueness and market equilibrium conditions for lossless-DLMPs with intertemporal energy requirements from FLs and DGs. However, lossless (DC) power flow formulation is used to represent distribution grids. As high resistive/impedance ratios exist in distribution grids, losses must also be considered in the proven market equilibrium conditions of lossless-DLMPs in [5]–[8]. As lossy-DLMPs are based on non-linear power flows to compute DLMPs, they are naturally more accurate in terms of representing grid conditions as compared to lossless-DLMPs. Recent works have considered linearized [10], [11], convexified [3], [12], and global power balance [1], [3], [13] variations of AC power flows to compute lossy-DLMPs. The linearized lossy-DLMP methods [10], [11] approximate AC power flows around a flat (1 p.u.) voltage magnitude across the distribution grid. Naturally, this inflicts errors in DLMPs at nodes farther away from the root-node, where a large voltage drop might occur. The convexified lossy-DLMP methods [3], [12] are more accurate than linearized AC power flows, however it does not translate to intuitive DLMP formulations [3]. This is important since the DSO must be able to interpret DLMPs and translate it into its financial settlements with market participants. The lossy-DLMPs in a most generic form are obtained from the global power balance formulation [1], [3], [13], allowing individual components of lossy-DLMPs to be analyzed in detail, which may help in interpreting DLMPs for distribution grids in a manner similar to the standardized LMP formulation of transmission grids [4]. However for DLMPs, the challenge in this formulation lies in expressing power flows which have higher nonlinearities in distribution grids as compared to those of transmission grids [3]. Another important feature of transmission grid energy markets is that the final LMP value is represented as the sum of its energy, loss and (line flow) congestion components. However, from the above mentioned lossy-DLMP formulations, only [3], [10], [11], [13] explicitly represent DLMP as the sum of its energy, loss and congestion component. Nevertheless, works in [3], [13] do not provide tractable formulation for nonlinear power flows and works in [10], [11] use static approximations (around 1 p.u. voltage) which may inflict errors in DLMP values. Hence, there exists a need to develop a DLMP model which expresses DLMP at the node as components of its energy, system loss and congestion contribution to the grid along with sufficient accuracy of nonlinear power flows. In contrast to market equilibrium studies on losslessDLMPs, investigations on lossy-DLMPs have mostly focused on incorporating nonlinearities in DLMPs and have not discussed market equilibrium conditions under the presence of both instantaneous and planned time horizon flexible resource
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dispatch. This is important as flexibility resources (instantaneous and planned horizon dispatch) are envisioned to be an integral part of future grids and there must exist local distribution grid markets which is economically efficient, i.e., maximizing the overall grid surplus as well as the surplus of individual flexible resources. B. Technical Contribution In this paper, we present a tractable calculation methodology of DLMPs using a classic global power balance formulation [1], [3], [13] accounting for non-linear power flows along with multi-period dispatch from FLs and DGs operation. To tackle nonlinearities of power flows, we build upon the recent works on AC power flow approximation for distribution grids [14] and extend it to include two key components relevant for DLMPs, i.e., line flows and system losses. We deploy a trust-region based solution methodology [15] which iteratively solves convex subproblems of the overall global power balance formulation. Particular for DLMPs, these convex subproblems have two favorable properties. First, a high solution robustness is achieved as many off-the-shelf solvers exist to solve convex programs efficiently. Second, convex subproblems contain sensitivities which are directly deployed in calculating and interpreting DLMPs. Through optimality conditions, we then show that the presented DLMP formulation: i) is decomposable into energy, congestion, loss and voltage components and exists in equilibrium with grid conditions and ii) under the assumption of rational flexible resources (maximizing their individual surplus), is able to form a day-ahead local distribution grid market which achieves efficient resource allocation. In this way, we propose a method which provides a generic DLMP formulation with tractable solution methodology, beneficial for the ongoing discussions on future distribution grid markets. Sec. II and III respectively present the system model and problem statement of this paper. The proposed methodology for calculating DLMPs along with its market equilibrium existence and flexibility resource allocation is presented in Sec. IV. The simulation setup and results follow in Sec. V. N OMENCLATURE Only the most fundamental variables are mentioned here as other variables in the text are their derivatives. N, R, C n, m ∈ N N∈N nt ∈ N Y ∈ CN×N vt ∈ CN vt ∈ RN θ t ∈ RN st ∈ CN f/t st ∈ Cm slt cl n pcl t /qt ∈ R dg dg pt /qt ∈ Rn pflt /qflt ∈ Rn cht ∈ Rn Dt ∈ Rn×n zt ∈ Rn x n Cx t (yt ) ∈ R ΠGrid ∈ Rn p /q t
t
sets of natural, real, complex numbers grid nodes (excluding root-node), grid lines all nodes (including root-node) N := n + 1, time steps planning horizon, T = {1, . . . , nt } nodal admittance matrix of the grid complex voltage vector (vt = vt ejθt ) voltage magnitude vector voltage angle vector complex power injection vector complex line flow vector from/to nodes total complex grid losses active/reactive power of Constant Loads (CLs) active/reactive power of DGs active/reactive power of FLs state of charge FLs time-coupled drain matrix of FLs disturbance experienced by FLs x x cost function for yx t ∈ {pt , qt } with x ∈ {g, fl} where g ∈ {0, dg} and 0 as the root-node supply grid cleared active/reactive power DLMP
where [·]t means the value at time step t. Scalars (x) are
lower case; vectors (x) are bold and lower case; matrices are bold and upper case (X). Complex quantities are underlined (Y) with cl 1> N qt − 1n (qt f+ 2 f 2 |st | ≤ (st )
+
qflt )
=
: λpt
qtl
: :
2 |stt |2 ≤ (st+ t )
vL− t
≤
vL t
≤
: vL+ t
:
λqt µ+ sft µ+ stt µ− , µ+ vL vL t t
(9c) (9d) (9e) (9f)
dg+ pdg− ≤ pdg t t ≤ pt
dg+ : µdg− (9g) pt , µpt
dg+ qdg− ≤ qdg t t ≤ qt
dg+ : µdg− (9h) qt , µqt
fl+ fl pfl− t ≤ pt ≤ pt
fl+ : µfl− pt , µpt
chfl− t
≤
chfl0
+
Dt pflt
− zt ≤
chfl+ t
:
fl+ µfl− cht , µcht
(9i) (9j)
for all t ∈ T = {1, . . . , nt }. Constraints (9b) and (9c) respectively present global active and reactive power balance of the distribution grid. From (8), we denote active power loss as plt and reactive power loss as qtl . Constraint (9i) manges dispatch capabilities of FLs within the allowable inter-temporal SOC constraints (9j). All DGs’ active/reactive power are constrained through (9g)/(9h), respectively. Apparent power flowing in distribution grid lines from/to ends of the lines are constrained through (9d)/(9e). The variables listed to the right of each constraint (behind colon) are their respective Lagrangian multipliers. The optimization problem in (9) is a classical global energy balance formulation for calculating DLMPs [3], [4]. However, unlike transmission grid’s single period, it deploys multi-period modeling as: • it allows for a more accurate description of flexible load modeling, because closer to the consumption (distribution grids), static aggregated load modeling of transmission grids might not be accurate [22]; • it aids in scheduling flexibility resources with intertemporal energy requirements; and
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it allows for a wide range of model applications and extensions (see Sec. IV-F). However, a great barrier for solving (9) in its given form comes from constraints (9b)–(9f), as they are highly nonlinear and non-convex and in general not easy to represent in a closedform tractable fashion [see (4)–(8)]. Hence, this renders the DLMP determination from (9) difficult as off-the-shelf solvers can not solve it in this form, reducing solution reliability and future practical realization of DLMPs. •
B. Constraints Linearization We seek the following approximations of non-convex parts of (9b)–(9f), L
v ˜ Lt ≈ ˆ at + Mvt sLt , ˆ t + Msft sinj |˜ sft |2 ≈ b t , |˜ stt |2 p˜lt q˜tl
IV. P ROPOSED M ETHODOLOGY In order to deal with the above stated barrier for solving (9), we propose to implement a linearized variant of (9) along with a trust-region update procedure. In principle, at each iteration, convex sub-problems of (9) are obtained. These convex subproblem are then iteratively solved and checked against the actual power flow solution to guarantee feasibility and improvement in the obtained solution of convex subproblems.
≈ ≈ ≈
(12a) (12b)
t ˆ ct + Mst sinj t , l inj p dˆt + M t st , l eˆt + Mqt sinj t ,
(12c) (12d) (12e)
ˆt , ˆ which, for given operating conditions (ˆ at , b ct , dˆt , eˆt ), are linear in the augmented active and reactive power injec| | | tions vector for the whole grid sinj t := (pt , qt ) and PQ L L| L| | nodes st := (pt , qt ) . 1) Linear Voltage Magnitude: For a given operation (ˆ vLt , ˆ sLt ), (12a) is obtained by first directly choosing vt ,θ t sensitivity terms from Mp which corresponds to voltage t ,qt L
vt ,θ t magnitude, i.e., Mvt := (Mp )n and then setting ˆ at := t ,qt L
A. Implicit Power Flow Linearization We now build upon upon the work in [20] to obtain a linear approximation of the implicit power flow representation (4) and extend it for the relevant DLMP calculations. From [20, Lemma 1], (4) is actually a smooth manifold, allowing us to attach a tangent plane to every operating point ˆt satisfying x ˆt in (4). Hence, for a given input xref t and state x the nonlinear manifold (4), we have the following first order approximation for the new solution state x ˜t : 02N f (˜ xt ) J(ˆ xt ) x ˜ − x ˆ , (10) ≈ + t t G g(˜ xt ) xref t
v ˆ Lt − Mvt ˆ sLt . 2) Linear Squared Line Flow: For squared line flows f “from” nodes expressed as |sft |2 := diag(sf∗ t )st , we proceed to obtain the desired sensitivities as follows:
= +
where the explicit tangent (Jacobian) matrix is J(ˆ xt ) =
h
=
i hdiag(Y ∗ v ˆ ∗t )i + hdiag(ˆ vt )iNhYi R(ˆ vt ), −I2N ∈ R2N×4N ,
represented using the following shorthands =
(1 − η) then /* good approx. */ δ(m + 1) = min(2δ(m), δmax ) ; else δ(m + 1) = δ(m); end Step 4: Evaluate Solution Progress ; if σ(m) > τ then m = m + 1; Accept the iteration, set new states as x ˆt (m) ; else Reject the iteration and repeat using the modified region; end end
The above definition of σ(m) represents the ratio between the cost improvement of approximated system to the actual one. A smaller value of σ(m) shows that the current approximation does not represent the actual system and hence the optimization region must be reduced. For a considerably higher value of σ(m), the linear approximation is accurate and the system can move to a new operating point. As a termination criteria, when the change in two consecutive LMOPF dispatch, i.e., |∆˜ sdisp sdisp sdisp t (m)|∞ := max|˜ t (m) − ˜ t (m − 1)| where |·| is the absolute operator, is below a certain threshold , Algorithm 1 terminates. For the choice of γ, , η, τ and δmax , interested readers are referred to [23]. 3) Solution Uniqueness: For the implicit function representation of (4) with input xref t , following sufficient condition (17) exists for a unique solution for a practical grid operation, i.e. v ˆ t within usual operating range [14, Theorem 1]: (vt0 )2 > 4lmax st ,
(17)
with the approximation v ˜ t in (10) bounded by [14, Collary 2], |˜ vt − v ˆt| ≤
4 li lmax s2t . (vt0 )3
(18)
The quantities li , lmax and st are the impedance path to node i from root-node, maximum impedance length of the grid and total load of the grid, respectively. Now assuming the dispatch ˆ sdisp t (m) at iteration m satisfies (17), then the approximated dispatch ˜ sdisp t (m) from (15) is unique. This is because the LMOPF is a QP (see Remark 2). Moving on, as the Algorithm 1, through trust-region ensures that each iteration’s imposed dispatch ˜ sdisp t (m) satisfies grid operations from (4), it maintains the power flow uniqueness for practical operating conditions (17). Given the feasibility of (4) and the solution uniqueness (17), Algorithm 1 terminates when improvement in dispatch ˜ sdisp t (m) stops, achieving a unique solution for the given trust-region valid in (4), i.e. ˜ sdisp sdisp t (m) = ˆ t (m).
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4) Formulation/Solution Benefits: The proposed method (Algorithm 1) handles non convexity in two steps; (i) AC power flow calculations (for feasibility checking) and (ii) convex subproblems optimization (maximizing social welfare). Both these steps have off-the-shelf software packages that can handle large-sized problems. In this way, the proposed method relies on mature technology, providing high solution reliability and robustness. Moreover, approximations of (12) are already embedded in the LMOPF (15), aiding in the interpretation and calculation of DLMPs, as demonstrated next in Proposition 1 of Sec. IV-D. Note that instead of handling non-convexity in two steps (proposed model), another option could have been to formulate convexified power flow radial grid models [24]–[27]. However, we show in Sec. VII-B that the convexified formulation might not translate to an intuitive DLMP formulation and interpretation.
dg dg+ dg dg+ µdg+ ) = 0, µdg− ) = 0, (20h) pt (pt − pt pt (−pt + pt dg dg+ dg dg+ µdg+ ) = 0, µdg− ) = 0, (20i) qt (qt − qt qt (−qt + qt fl+ fl+ fl fl− fl µfl+ pt (pt − pt ) = 0, µpt (−pt + pt ) = 0, fl fl+ fl µfl+ cht (ch0 + Dt pt − zt − cht ) = 0, fl fl− fl µfl− cht (−ch0 − Dt pt + zt + cht ) = 0,
(20j) (20k) (20l)
along with all primal feasible conditions of the LMOPF and non-negative Lagrange multipliers. For active powers, only stationary condition (20b) contains root-node, which due to the implicit definition (see Sec. IV-B) turns all the root-node entries in M[·] to zero, leading to c0p,t = −λpt .
(21)
Now let respective marginal values of DGs and FLs be, t∈T \|T |
D. DLMPs: Decomposition & Market Equilibrium Utilizing the solution methodology provided in the above subsection, we now derive DLMPs, satisfying market equilibrium along with decomposable in grid conditions. Proposition 1: For active power procurement at each time step t, there exist active power DLMPs, i.e., ΠGrid pt , which are under equilibrium and completely representable for the whole grid as the sum of their (i) energy ΠEpt , (ii) loss ΠLpt , (iii) congestion ΠCpt and (iv) voltage ΠV pt components, i.e., E L C V n ΠGrid pt = Πpt + Πpt + Πpt + Πpt ∈ R .
uniquely determined as: ΠEpt | plt pt n | vLt pt n
−(M )
c0p,t
−
qtl pt
:=
|
(M )n c0q,t , ΠCpt vLt − ΠV pt := Mpt (µvLt −
(19)
c0p,t 1n , ΠLpt :=
:=
st | (Mptt )n µsft
+
µ+ )2 . vLt
(M ) µstt and Proof: Consider the KKT conditions to be satisfied by the solution of the LMOPF problem ∀t ∈ T = {1, . . . , nt }3 : l
|
l
|
sf
|
cflp,t + λpt 1n − (Mpptt )n λpt − (Mqptt )n λqt + (Mptt )n µsft |
st
L
fl− +(Mptt )n µstt + (Mvptt )| (µ− − µ+ ) + µfl+ pt − µpt vL vL t
t
t∈T \|T |
}| { z fl− + Dt (µfl+ cht − µcht ) = 0,
(20a)
| l | l | sf − λpt 1N + Mpptt λpt + Mqptt λqt − Mptt µsft st | | + dg+ dg− −Mptt µstt − Mvptt (µ− vt − µvt ) − µpt + µpt = | l | l | sf cgq,t − λqt 1N + Mpqtt λpt + Mqqtt λqt − Mqtt µsft st | | + dg+ dg− −Mqtt µstt − Mvqtt (µ− vt − µvt ) − µqt + µqt = g > cl fl −λpt (1> ˜lt ) = 0, N pt − 1n (pt + pt ) − p g > cl fl −λqt (1> ˜tl ) = 0, N qt − 1n (qt + qt ) − q f 2 2 2 µsft (|˜ st | − (sf+ stt |2 − (st+ t ) ) = 0, µsft (|˜ t ) ) = 0, − µ+ (˜ vLt − vL+ vLt + vL− t ) = 0, µvLt (−˜ t ) = 0, vLt
cgp,t
[·]
0, (20b)
0, (20c) (20d) (20e) (20f) (20g)
Mp /q corresponds to active/reactive power parts of M[·] . t t Marginal cost of active/reactive power supply at the root-node is c0p,t /c0q,t . 3 As stationary condition (20a) contains inter-temporal constraints, note that the last term only exists for non-terminal period, i.e., T \|T |, else its zero [7]. −/+ −/+ | | Also in (20b), (20c) µvt := (0, µ L ) as root-node voltage is fixed. 2 Matrix
vt
ΠFlex pflt Flex Πpdg t
:= :=
}| { fl− − µch ) ∈ Rn t
z
fl+ fl− −cflp,t − µfl+ pt + µpt − Dt (µcht dg+ dg− n cdg p,t + µpt − µpt ∈ R
(22)
As the definitions in (22) represent internal constraints of the flexibility resources, the cleared DLMPs ΠGrid pt , which is a grid equilibrium, must satisfy the following condition: Grid Flex ΠFlex pfl = Πpt = Πpdg . t
(23)
t
After substituting (21) in [(20a), (20b)] and then utilizing (23), we recover the individual components of ΠGrid pt , as shown in (19). Moreover, from (23) these components exists in equilibrium. Hence, Proposition 1 holds true. Similar to active power DLMPs ΠGrid pt , reactive power DLMPs ΠGrid can be derived using a similar method, i.e. qt evaluating the marginal cost of supplying reactive power at the root-node, c0q,t with its corresponding stationary condition of (20c). However, whether this marginal cost of supplying reactive power at the root-node is made available by existing wholesale markets is still a topic of ongoing discussion [1], [17], [18]. Nevertheless, the presented model is flexible enough to provide an option of pricing reactive power only as a function of its marginal value at the root-node, i.e. c0q,t . If this information is not available at the root-node, this price can simply be set to zero. E. DLMPs: Day-ahead Local Distribution Grid Market We show in this subsection that the proposed DLMP model can be utilized by the DSO to operate a day-ahead local distribution grid market as: 1) The DSO forecasts its daily underlying grid demand along with the cost to supply it using the marginal-cost at the root-node, i.e. c0p,t /c0q,t ; 2) DGs/FLs submit their day-ahead instantaneous/multitemporal bids, energy requirements and dispatch capabilities; and 3) The DSO clears DLMPs for each interval and pass them to DGs/FLs. Now we show that this local market is able to achieve efficient flexible resource allocation. This means that the cleared DLMPs of the local market in (23), optimizing the overall
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system dispatch ˆ sdisp in (15), achieves exactly same dispatch t as to when DGs/FLs maximize their individual surpluses using their corresponding individual DLMPs. The following proposition shows this property. Proposition 2: Under the assumption of no forecast error, the DSO cleared DLMPs ΠGrid pt correspond to a unique solution, maximizing the overall social welfare (15) as well as FLs’ individual surplus4 . Proof: Consider individual FLs’ maximization problem X
max
| fl −(ΠGrid pt ) pt
t∈T fl+ fl pfl− t ≤ pt ≤ pt fl fl chfl− t ≤ ch0 + Dt pt
s.t.
(24a) fl+ : µfl− pt , µpt
− zt ≤
fl− fl+ chfl+ t : µcht , µcht
(24b) (24c)
∀t ∈ T = {1, . . . , nt }. As LMOPF (15) has a strictly convex cost function (Remark 2), the inclusion of DLMPs (linear in pflt ) also makes the cost function of individual FL problem (24a) strictly convex. Combining this with affine constraints (24b), (24c), the individual FLs problem (24) is then also a QP, with a unique minimizer (ˆ pflt )ind satisfying its following necessary and sufficient KKT conditions (25): t∈T \|T |
z
}|
{
fl+ fl− fl+ fl− ΠGrid pt + µpt − µpt + Dt (µcht − µcht ) = 0, fl+ fl+ fl fl− fl µfl+ pt (pt − pt ) = 0, µpt (−pt + pt ) = 0, fl+ fl fl µfl+ cht (ch0 + Dt pt − zt − cht ) = 0, fl− fl fl µfl− cht (−ch0 − Dt pt + zt + cht ) = 0,
(25)
along with the primal feasible (24b), (24c) and all Lagrange multipliers non-negative constraints. Observe that the grid cleared DLMPs, ΠGrid pt , deployed in the individual problems of (25) are in equilibrium with ΠFlex in (22). This makes all individual KKT conditions pflt of (25), in fact, a part of the overall KKT conditions (20), meaning that a valid overall solution (ˆ pflt )all , satisfying (20), is also a solution to (25). On the other hand, the individual problem (24) does not contain any grid constraints, so a valid individual solution (ˆ pflt )ind to (25) may not necessarily satisfy (20). However, ΠGrid shared among the individual pt and overall problem is obtained around a unique operating point, due to strictly convex subproblem (see Sec. IV-C3), making the overall solution (ˆ pflt )all of (20) unique. Moreover, as the individual FL problem (24) is also a strictly convex problem, it also constitutes a unique solution (ˆ pflt )ind . Due to this uniqueness, both the individual problem solution and the overall solution are similar, i.e. (ˆ pflt )ind = (ˆ pflt )all . F. Discussion Practical Implication: The local distribution grid market proposal through the proposed method in Sec. IV-E can be simply extended to co-exist with the current wholesale dayahead energy markets [28]. This can be achieved by first FLs/DGs purchasing/selling energy from the wholesale market. The schedules of FLs/DGs are then submitted to the DSO. The DSO can then proceed to run its local market (steps 1-3 4 Similar
proposition and subsequent proof exists for DGs.
Fig. 2. The modified 33-bus distribution grid with two FLs and two DGs, with PSP (root-node) connection from the transmission grid. For fully exploring the proposed method FL-1 and DG-1 are placed closer to the root-node, whereas FL-2 and DG-2 are placed far from the root-node.
of Sec. IV-E) and formulate DLMPs, which are then passed on to FLs/DGs. FLs/DGs can then include these DLMPs into their individual energy planning and proceed with their participation in the day-ahead wholesale market. Application Support: A possible application of the proposed day-ahead local market, described in Sec. IV-E, is in developing grid-friendly demand response and congestion management programs. This is possible because the cleared DLMP at a particular node (23) automatically reflects its contribution of delivered energy towards system losses, congestion and voltage violations. Hence, the submitted schedules by FLs/DGs can be evaluated by the DSO and eventually can lean towards more grid-friendly demand response and congestion management schemes. Note that for lossless DLMPs, similar model applications have been proposed in [5]–[8]. Formulation Extensions: Since the proposed model in (15) represents flexibility resources for both instantaneous and multi-period dispatch, this potentially allows for multiple formulation extensions. For example, multi-period FLs formulation in (15) can naturally be extended to include wide range of generic inter-temporal energy-constrained flexibility resources, such as energy storage systems and DGs with rampup/down limitations [1], [2]. V. S IMULATION S ETUP & R ESULTS The proposed method is tested on the IEEE 33-bus distribution system [16]. The modified configuration of the grid is shown in Fig. 2. A realistic scenario with DGs and FLs operating under different cost functions and dispatch capabilities are simulated as follows; (1) the active and reactive power dispatch for DGs are within range [0, 0.5] MW and [−0.3, 0.3] MVar, with marginal cost of active and reactive power set at 10 $/MWh and 3 $/MVarh for the whole day and (2) the active power dispatch for FLs is within the range [−1.47, 0] MW (derived from [5]), to be optimized using the day-ahead cleared price at the root-node (taken from [5, Fig. 6]). A small value of price sensitivity coefficient of 1e-4 $/MWh2 (MVarh2 ) is chosen to keep all cost functions strictly convex. This value is in line with the price sensitivity coefficients assumed in [7]5 . More information on obtaining these coefficients is provided in [8], [9]. Parameters of the trust-region in Algorithm 1 are taken from [23]. All simulations are carried out on a 2.4GHz processor with 64-GB RAM. Optimization problems are 5 A considerably large value of these coefficients makes flexible resources to dispatch their schedule more conservatively, as then the quadratic term (Hessian) dominates their respective cost functions (see Definition 1).
8 0.5
solved using YALMIP [29] with GUROBI [30] as a solver. Power flows for the trust-region evaluation steps in Algorithm 1 are performed using MATPOWER [21].
0 −0.5 −1
A. System Dispatch using DLMPs
20
15
10
0
5
ΠGrid pt ($/MWh)
c0p,t
10
15
(ΠGrid pt )18
20
25
30
(ΠGrid pt )22
35
40
(ΠGrid pt )25
45 (ΠGrid pt )33
20
pfl2 t
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
45
0 −0.5 −1
0.5 0 −0.5 −1
Time steps (half hour)
Fig. 4. Active power dispatch values for Scenario 1 (top), 2 (middle) and 3 (bottom) for all flexible nodes. TABLE I DLMP S FOR SCENARIO 1 ( TOP ), 2 ( MIDDLE ) AND 3 ( BOTTOM ) AT TIME STEP t = 10 AND NODE i. ΠEpt
ΠLpt
dg/fl
ΠC pt
ΠV pt
ΠGrid pt cp,t
ΠM pt
ΠD pt
ΠFlex dg/fl
18 1.62 22 -0.04 25 11.52 1.53 33 4.10
0 0 0 0
0 0 0 0
13.14 11.48 10 13.06 15.63 -11.52
3.14 1.48 -2.84 -1.41
– – 27.42 28.56
13.14 11.48 13.06 15.63
18 1.45 22 -0.04 11.52 25 1.48 33 3.50
0 0 0 0
0.83 -0.04 0.24 2.78
13.80 11.44 10 13.25 17.80 -11.52
3.80 1.44 -2.67 0
– – 27.44 29.32
13.80 11.44 13.25 17.80
18 1.42 22 -0.05 11.52 25 1.26 33 3.50
0 0 0.23 0
0.82 -0.04 3.25 2.76
13.77 11.42 10 16.26 17.79 -11.52
3.77 1.42 0 0
– – 27.79 29.32
13.77 11.42 16.26 17.79
i
pt
For time step 10 and all scenarios, we present market fl+/dg+ fl−/dg− equilibrium in Table 1. In Table 1, ΠM −µpt , pt = µpt D fl+ fl− and Πpt = Dt (µcht − µcht ). Shown in bold in Table 1, it can be observed that flexible DLMPs representing the internal flexible node dynamics exist in equilibrium with the grid Flex cleared DLMPs, i.e., ΠGrid pt = Πpdg/fl . B. Model Comparisons
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
45
20
15
10
pfl1 t
t
15
10
pdg1 t
0.5
MW
To explore the proposed method in detail, we simulate three scenarios with the resultant active power DLMPs along with dispatch quantities presented in Fig. 3 and Fig. 4, respectively. Scenario 1 is simulated by relaxing all nodal voltages and line flow constraints, i.e., to only show the effect of losses in DLMPs. As dispatching energy from DGs is cheaper than supplying from the root-node, i.e. 10 $/MWh < c0p,t , ∀t ∈ T , they are fully dispatched for the whole time horizon in this scenario. However, this does not imply that nodes with DGs also experience lower DLMPs. This is because whenever DGs are dispatch fully, the extra MW amount comes from the higher marginal cost providing the root-node. At step 10, FLs draw a large amount of energy, raising their DLMP levels in proportion to their individual contributions towards the overall losses to the grid. Scenario 2 constrains all nodal voltage magnitude between 0.92 and 1.05 p.u., causing an increase in DLMPs at time step 10 for FL2 as higher loading at node 33 binds lower voltage limits. As a consequence, FL2 draws lesser power at time step 10, as compared to Scenario 1. The same dispatch phenomenon is observed for DG1 (node 22), where active power dispatch reduces in order to adhere to its respective voltage limits. This causes DLMPs at node 22, now fully served by DG1 to be equal to its marginal cost, i.e., 10 $/MWh, except at time step 10, where higher grid loading from FLs, allows DGs to dispatch fully, causing the extra MW amount to flow from the root-node and raising DLMP levels. Scenario 3 constrains the distribution line serving FL1 by 1.6 MVA. As compared to Scenario 1 & 2, this reduces the power drawn by FL1 at time step 10 and increases its DLMP value due to binding line flows.
pdg2 t
Time steps (half hour)
Fig. 3. Active power DLMPs for Scenario 1 (top), 2 (middle) and 3 (bottom) for all flexible nodes along with the marginal cost at the root-node c0p,t .
In Fig. 5, the performance of the proposed LMOPF model is compared against a state-of-the-art linearized power flow model (LIOPF) [10] and an interior point MATPOWER model (ACOPF) [21]. As LIOPF and MATPOWER only provide DLMPs for a single time step (instantaneous dispatch), we also adopt our proposed model for single time step. The dispatch capabilities of DGs/FLs remain the same as during the previous scenarios, however we simulate two cases to fully explore the proposed model. Case-1: For all flexible nodes and the root-node, this case fixes the active and reactive power marginal costs at 10 $/MWh and 3 $/MVarh. It can be observed in Table 2 that the proposed
9
12
14
11
12
LIOPF
LMOPF
11
10 0
10
20
30
ACOPF
10
10 0 0
10
LIOPF
20
30
LMOPF
4.5
ΠGrid ($/MVarh) q
ACOPF
12
ΠGrid ($/MWh) p
ΠGrid ($/MWh) p
13
4
6
10
20 Nodes
30
0
10
20 Nodes
30
80
100
120
140
12 11 10 0
100
200
0
200
400
300
400
500
600
800
1,000
12 11
Fig. 5. Active (top) and reactive power (bottom) DLMPs of ACOPF [21], LIOPF [10] and the proposed LMOPF methods for case-1 (left) and case2 (right).
LMOPF method achieves almost identical dispatch quantities and DLMPs to the non-convex ACOPF solution. Moreover, the proposed LMOPF, utilizing trust-region iterations, provides an improved solution quality compared to the LIOPF method. This is because for nodes far from the root-node large voltage drops exist causing higher errors in linearization, which is performed around a flat nodal voltage profile (1 p.u) and negligible angle difference. As FLs, due to their locations, with same marginal values to DGs are not dispatched, next we present Case-2 to evaluate their presence on DLMPs. Case-2: In this case, marginal utility for all FLs are increased to 15 $/MWh, all remaining settings are similar to Case-1. The proposed LMOPF method again achieves almost identical dispatch quantities and DLMPs as compared to the non-convex ACOPF solution [21]. The large power drawn by the FLs now binds the lower voltage and line flow limit at node 25 and 33. TABLE II D ISPATCH Q UANTITY C OMPARISON
Units
60
13
3 0
40
13
3.5 4
20
Case-1
Case-2 ACOPF LMOPF LIOPF 0.50 0.50 0.50 0.30 0.30 0.30
DG1
MW MVar
ACOPF LMOPF LIOPF 0.50 0.50 0.50 0.30 0.30 0.30
DG2
MW MVar
0.28 0.13
0.29 0.13
0 0
0.39 0.12
0.39 0.12
0.50 0.17
FL1 MW
0
0
0
-1.16
-1.10
-1.09
FL2 MW
0
0
0
-1.24
-1.25
-1.36
The general observation from both cases is that for nodes 19-22, similar DLMPs are obtained from all three methods. This happens because voltage magnitudes are close to their upper voltage limits and binding at node 22, making its feed-in branch experiencing DLMPs at its marginal cost, i.e. 10 $/MWh and 3 $/MVarh. For Case-2, this effect is also caused by line flow constraint binding at node 33, causing the respective DLMPs to be closer to FLs’ marginal utility value, i.e. 15 $/MWh. It was also confirmed that, for both cases, regardless of the initialization point, the proposed LMOPF
10 Nodes
Fig. 6. Active power DLMP comparison between the proposed LMOPF, ACOPF [21] and LIOPF [10] for 141-nodes (top), 564-nodes (middle) and 1128-nodes (bottom) grid.
method always converged to the same dispatch and consequently DLMPs (solution uniqueness) within only 4 iterations. C. Model Scalability To show the effectiveness of the proposed model, we implement the proposed model on three distribution grids containing 141-nodes [31], 564-nodes and 1128-nodes. For all the grids, the marginal cost settings for DGs/FLs are kept similar to those of Case-2 of Sec. V-B. We make the 564-nodes and the 1128-nodes grid by duplicating the 141-nodes grid, while preserving its original root-node [10]. Similarly, the number of DGs/FLs in the 564-nodes grid and the 1128-nodes grid are proportionally increased as compared to 141-nodes grid. In this way, we not only evaluate the feasible solution projection step but also the trust-region minimization step of the proposed model (See Algorithm 1). The comparison of the proposed model with other methods in the literature is also shown in Fig. 6. We only present active power DLMPs in Fig. 6 as reactive power DLMPs show similar results and are left out here to save space. Similar to Fig. 5, Fig. 6 reiterates that the proposed model achieves the same results as MATPOWER’s interior point ACOPF model, while outperforming the state-of-the-art linearized model [10]. It is to be noted that both the interior point and semidefinite programming ACOPF solvers of MATPOWER produced exactly the same result. Table. III details the information regarding the simulated grids and the resultant computation efforts of the proposed model. It can be seen that the proposed model scales well with the increase in the size of the grid and the number of flexibility resources (DGs/FLs). VI. C ONCLUSION & F UTURE W ORKS We proposed a new distribution locational marginal price (DLMP) model which utilized the classical global energy balance formulation and combined with the recent works
10
TABLE III P ROPOSED M ODEL S CALABILITY Nodes
DGs
FLs
Iteration
Time (sec.)
141
2
2
3
1.5
564
8
8
3
3.3
1128
16
16
4
13.3
and cost function given in (7) and Sec. VII-A. For more information regarding the formulation, interested readers are referred to [26]. Constraints (26b)-(26e) represent AC power flow relaxation. The constraint (26e) is actually a second order cone constraint, if satisfied as an equality constraint makes the relaxation exact [27]. For detailed information on SOCP, interested readers are referred to [26], [27]. max
on AC power flow linearization. Relevant for DLMPs, we extended the AC power flow linearizations to represent grid losses and line flows. The developed model showed that it is able to depict market equilibrium conditions and efficiently allocates flexible resources in a day-ahead local market. Moreover the calculated DLMPs are decomposable into their most generic grid components of energy, loss, congestion and voltage. To show the efficacy of the proposed model, we implemented on a benchmark distribution systems and compared it with other state-of-the-art methods. An interesting future works exists in extending the proposed model to include various energy storage systems and DGs with ramp-up/down limitations; generic multiphase and unbalanced distribution grids; and voltage, current and impedance dependent load models. VII. A PPENDIX A. Cost/Utility Functions Let λ be the cleared active power per unit price, then the cost/utility function for DGs/FLs in (1) follows: (
(
)) ∂Uflt (pflt )) = λ , ∂pflt pflt ∈Rn ( ( )) g g g g g g g ∂Ct pt pt = argmax λpt − Ct (pt ) = max 0, pt | = λ . g g ∂pt p ∈Rn
pflt = argmax Uflt (pflt ) − λpflt = max
0,
pflt |
t
Same interpretation holds for reactive power cost functions. node 0
f
2
|v 0|
node 1
r 0 , x0
r 1 , x1
node 2
node n
f
f f |i 0|2, p0, q0 | i 1|2, p1, q1 |v 1|2 , p1, q1 |v 2|2, p2,q2
|v n|2, pn, qn
Fig. 7. Notations used to represent branch flow convexified power flow model of the radial grid [26]. For line connecting node j and k, squared line current, active power flow, reactive power flow, resistance and reactance are represented as: isq := |ij |2 , pfj , qjf , rj and xj , respectively. For j node j, squared voltage, active power injection and reactive power injection are represented as: vjsq := |vj |2 , pj and qj , respectively.
B. Convexified DLMP Formulation In order to aid in developing branch flow convexification method [16], [26], [27], we adopt scalar notation here to represent a radial grid in Fig. 7. We formulate convexified ACOPF for a generic injections (pj , qj ) at each node j and single time-step in (26)6 . In (26), sfj and cpj (·)/cqj (·) are respectively the scalar version of the line flow “from” 6 The full formulation (analogous to (9)) which considers FLs/DGs along with inter-temporal energy and actuator constraints is a straight forward procedure and is left out here for exposition simplicity and brevity.
−
n X
cpj (pj ) + cqj (qj )
(26a)
j=1
s.t. pfj = pfj+1 + rj isq j − pj+1 f qjf = qi+1 + xj isq j − qj+1 sq sq vj = vj+1 + 2(rj pfj + xj qif ) (pfj )2 + (qjf )2 ≤ isq j vjsq (pfj )2 + (qjf )2 ≤ (sfj + )2 vjsq− ≤ vjsq ≤ vjsq+
: λpj
(26b)
λqj λvj
(26c) (26d)
: µij
(26e)
: µsj
(26f)
: −
(rj2
+
x2j )isq j
:
:
v+ µv− j , µj
∀j ∈ {0, . . . , n − 1}
(26g)
Now considering an exact solution of (26), the following KKT conditions are then satisfied: cpj + λpj = 0 cqj + λqj = 0
(27a) (27b)
λpj−1 = λpj + 2rj λvj − 2pfj (βj + µsj ) = 0
(27c)
λqj−1 = λqj + 2xj λvj − 2qjf (βj + µsj ) = 0 rj λpj + xj λqj − (rj2 + x2j ) + µij = 0 (pfj )2 + (qjf )2 v− v v i + µv+ µj j − µj + λj − λj−1 (vjsq )2
(27d) (27e) = 0 (27f)
∀j ∈ {1, . . . , n} Along with the primal feasible (26) and non-negative Lagrange µi multipliers conditions. In (27), we have βj := vsqj . j For active power, consider DLMP at node j to be defined p as λj . Then from (27c), we have DLMP at node (j) λpj dependent upon DLMP at its ancestor-node (j − 1) λpj−1 , along with other terms which are only dependent on the node (j) 2rj λvj and the line connecting the node (j) and its ancestor-node (j − 1) 2pfj (βj + µsj ). Same explanation holds for the reactive power DLMP at node (j), i.e., λqj . 1) DLMP Representation Issues: The DLMP value obtained from the above mentioned convexified formulation (27) might be interpreted as follows: a marginal change in injection at node (j) only requires a marginal change of injections at its own node (j) and its ancestor-node (j − 1), while leaving reset of the grid unaffected. However, this interpretation is not physically true. As it has been shown in [3, Proposition 3.3] that a marginal change in the injection at node j also affects variables (e.g. voltages, line flows) at nodes other than its ancestor-node (j − 1). Moreover, Lagrange multipliers λvj and µij do not have straight forward interpretation. In light of these representation issues, we adopt global power balance formulation and its trust-region based iterative linearization for calculating DLMPs.
11
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